弯曲绝对L-无穷代数的积分理论
The integration theory of curved absolute L-infinity algebras
AI总结：
本文引入弯曲绝对$\mathcal{L}_\infty$-代数，通过完全Bar构造和内在模型范畴结构发展其积分理论，并应用于有理同伦理论与形变理论，为有限型幂零空间提供有理模型并编码形变复形与形式几何。
摘要：
在本文中，我们引入了弯曲绝对$\mathcal{L}_\infty$-代数的概念，这是一种表现得像弯曲$\mathcal{L}_\infty$-代数的结构，其中所有运算的无穷和根据定义都是良定义的。我们通过在积分理论中引入两种新方法来发展它们的积分理论：完全Bar构造和内在模型范畴结构。这使得我们能够从概念的角度快速推广该理论的所有基本结果。我们提供了该理论在有理同伦理论中的应用，并表明弯曲绝对$\mathcal{L}_\infty$-代数为有限型幂零空间提供了有理模型，而无需任何带基点或连通的假设。此外，我们证明了有理空间的同调可以作为完全Bar构造的同调来恢复。我们还为有理映射空间构造了新的更小模型，而对源单纯集没有任何假设。另一个应用来源是形变理论：在代数方面，我们证明了为了编码（余）代数的$\infty$-态射的形变复形，弯曲绝对$\mathcal{L}_\infty$-代数是必不可少的。在几何方面，我们由导出仿射栈构造了一个弯曲绝对$\mathcal{L}_\infty$-代数，并表明它编码了存在于基域的任何有限域扩张中的任意有限点集的形式几何。
英文摘要：
In this article, we introduce the notion of a curved absolute $\mathcal{L}_\infty$-algebra, a structure that behaves like a curved $\mathcal{L}_\infty$-algebra where all infinite sums of operations are well-defined by definition. We develop their integration theory by introducing two new methods in integration theory: the complete Bar construction and intrinsic model category structures. They allow us to generalize all essential results of this theory quickly and from a conceptual point of view. We provide applications of our theory to rational homotopy theory, and show that curved absolute $\mathcal{L}_\infty$-algebras provide us with rational models for finite type nilpotent spaces without any pointed or connected assumptions. Furthermore, we show that the homology of rational spaces can be recovered as the homology of the complete Bar construction. We also construct new smaller models for rational mapping spaces without any hypothesis on the source simplicial set. Another source of applications is deformation theory: on the algebraic side, we show that curved absolute $\mathcal{L}_\infty$-algebras are mandatory in order to encode the deformation complexes of $\infty$-morphisms of (co)-algebras. On the geometrical side, we construct a curved absolute $\mathcal{L}_\infty$-algebra from a derived affine stack and show that it encodes the formal geometry of any finite collection of points living in any finite field extension of the base field.
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